Pure resolutions, linear codes, and Betti numbers

Journal of Pure and Applied Algebra 224 (2020) 106385

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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa

Pure resolutions, linear codes, and Betti numbers Sudhir R. Ghorpade a,∗,1 , Prasant Singh b,2 a

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Department of Mathematics and Statistics, UiT The Arctic University of Norway, N-9037 Tromsø, Norway b

a r t i c l e

i n f o

Article history: Received 25 October 2019 Received in revised form 23 March 2020 Available online 3 April 2020 Communicated by S. Iyengar MSC: 13D02; 94B05; 05B25; 51E20

a b s t r a c t We consider the minimal free resolutions of Stanley-Reisner rings associated to linear codes and give an intrinsic characterization of linear codes having a pure resolution. We use this characterization to quickly deduce the minimal free resolutions of Stanley-Reisner rings associated to MDS codes as well as constant weight codes. We also deduce that the minimal free resolutions of Stanley-Reisner rings of first order Reed-Muller codes are pure, and explicitly describe the Betti numbers. Further, we show that in the case of higher order Reed-Muller codes, the minimal free resolutions are almost always not pure. The nature of the minimal free resolution of StanleyReisner rings corresponding to several classes of two-weight codes, besides the first order Reed-Muller codes, is also determined. © 2020 Elsevier B.V. All rights reserved.

1. Introduction One of the interesting developments in algebraic coding theory in the recent past is the association of a fine set of invariants, called Betti numbers, to linear error correcting codes. This is due to Johnsen and Verdure [16] and their idea is as follows. Some basic terminology used below is reviewed in the next section. Let C be a q-ary linear code of length n and dimension k and let H be a parity check matrix of C. The vector matroid corresponding to H is a pure simplicial complex, say Δ, and its Stanley-Reisner ring RΔ over Fq is a finitely generated standard graded Fq -algebra of dimension n − k. As such it has a minimal graded free resolution. Moreover, Δ is shellable, thanks to a classical result that goes back to Provan [22] (see also Björner [4, §7.3]). Hence RΔ is Cohen-Macaulay. (See, for example, [13, Ch. 6, §2].) So by the * Corresponding author. E-mail addresses: [email protected] (S.R. Ghorpade), [email protected] (P. Singh). Sudhir Ghorpade is partially supported by DST-RCN grant INT/NOR/RCN/ICT/P-03/2018 from the Dept. of Science & Technology, Govt. of India, MATRICS grant MTR/2018/000369 from the Science & Engg. Research Board, and IRCC award grant 12IRAWD009 from IIT Bombay. 2 Prasant Singh is supported by a postdoctoral fellowship from grant 280731 from the Research Council of Norway. 1

https://doi.org/10.1016/j.jpaa.2020.106385 0022-4049/© 2020 Elsevier B.V. All rights reserved.

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Auslander-Buchsbaum formula, the length of any minimal free resolution of RΔ is n − (n − k) = k, and it looks like Fk −→Fk−1 −→ · · · −→ F1 −→ F0 −→ RΔ −→ 0

(1)

where F0 = R := Fq [X1 , . . . , Xn ] and each Fi is a graded free R-module of the form Fi =



R(−j)βi,j

for i = 0, 1, . . . , k.

(2)

j∈Z

The nonnegative integers βi,j thus obtained depend only on C (and not on the choice of H or the minimal free resolution of RΔ ), and are the Betti numbers of C. Thus we may refer to (1) as a (graded minimal free) resolution of C. Such a resolution is said to be pure of type (d0 , d1 , . . . , dk ) if for each i = 0, 1, . . . , k, the Betti number βi,j is nonzero if and only if j = di . If, in addition, d0 , d1 , . . . , dk are consecutive, then the resolution is said to be linear. Johnsen and Verdure [16] showed that the Betti numbers of a code C contain information about all the generalized Hamming weights di (C) of C. In fact, they showed that di (C) = min{j : βi,j = 0} for i = 1, . . . , k.

(3)

More recent work of Johnsen, Roksvold and Verdure [18] shows that the Betti numbers of C and its elongations determine the so-called generalized weight polynomial of C. Thus, if we combine this with the results of Jurrius and Pellikaan [19], then we obtain a direct relation between the generalized weight enumerator of C and the Betti numbers of C and of its elongations. It is clear therefore that explicit determination of Betti numbers of codes would be useful and interesting. On the other hand, it is usually a hard problem, except in some special cases. The simplest class of codes for which Betti numbers are completely determined is that of MDS codes where the minimal free resolution is linear. The next case is that of simplex codes or dual Hamming codes, which are essentially the prototype of constant weight codes (indeed, by a classical result of Bonisoli [6], every constant weight code is a concatenation of simplex codes, possibly with added 0-coordinates). For such codes, the Betti numbers were explicitly determined by Johnsen and Verdure in another paper [17]. In this case, it turns out that the resolution is pure, although not necessarily linear. In general, Betti numbers of pure resolutions are relatively easy to determine, thanks to a formula of Herzog and Kühl [14], which in the case of linear codes provides an expression for the Betti numbers in terms of the generalized Hamming weights. So the result for simplex codes can be deduced from it if one knows that their (minimal free) resolutions are necessarily pure. Partly with this in view, we consider the question of obtaining an intrinsic characterization for a linear code to have a pure resolution. A complete characterization is given in Theorem 3.6. This is then applied to show that the first order Reed-Muller codes have a pure resolution and all their Betti numbers can be described explicitly. On the other hand, we show that Reed-Muller codes of order 2 or more do not, in general, have a pure resolution. As a corollary, it is seen that the property of admitting a pure resolution is not preserved when passing to the dual. The first order Reed-Muller codes are examples of two-weight codes, and it is natural to ask if a similar result holds for every two-weight code. However, unlike constant weight codes, the structure of two-weight codes is far more complicated and it is a topic of considerable research in coding theory and finite projective geometry. We refer to the survey of Calderbank and Kantor [8] and the references therein for a variety of examples of two-weight codes. We also take up the question of determining the Betti numbers of many of these codes. It is seen that the resolution is not always pure and thus we can not appeal to the Herzog-Kühl formula. Nonetheless, we succeed in determining the Betti numbers of many two-weight codes, partly by using a set of equations due to Boij and Söderberg [5]. It appears that the technique of Boij-Söderberg

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equations used here could be fruitful in the determination of Betti numbers of many important classes of linear codes. We remark that although our results on the Betti numbers of simplex and first order Reed-Muller codes using the Herzog-Kühl formula were obtained independently in early 2015, Trygve Johnsen [15] has informed us that similar formulas are obtained in the Ph.D. thesis of Armenoff [1] and the Master’s thesis of Karpova [21]. In any case, our emphasis here is on the general characterization of purity, applications to Reed-Muller codes that are not only of first order, but also of higher order, and the determination of Betti numbers of many two-weight codes. 2. Preliminaries Fix, throughout this paper, positive integers n, k with k ≤ n and a finite field Fq with q elements. We denote by [n] the set {1, . . . , n} of first n positive integers. Also, 2[n] denotes the set of all subsets of [n]. For any finite set σ, we denote by |σ| the cardinality of σ. By a [n, k]q -code, we shall mean a q-ary linear code of length n and dimension k, i.e., a k-dimensional subspace of Fqn . 2.1. Codes and matroids Let C be a [n, k]q -code and let H be a parity check matrix of C. For i ∈ [n], let Hi denote the i-th column of H. Define Δ := {σ ∈ 2[n] : {Hi : i ∈ σ} is linearly independent over Fq }. The ordered pair ([n], Δ) is a matroid, and we call it the matroid associated to the code C. Elements of Δ are called independent sets of this matroid. A maximal independent set in Δ is called a basis of the matroid. It is well-known that every basis of a matroid has the same cardinality and this number is called the rank of the matroid. If σ ⊆ [n] and if we let Δ|σ := {τ ∈ Δ : τ ⊆ σ}, then (σ, Δ|σ) is a matroid, called the restriction of the matroid ([n], Δ) to σ; the rank of this restricted matroid is called the rank of σ and denoted by r(σ); the difference |σ| − r(σ) is denoted by η(σ) and called the nullity of σ. Evidently, the rank of the matroid ([n], Δ) is the rank of H, which is n − k, and so the nullity of any σ ⊆ [n] ranges from 0 to k. For 0 ≤ i ≤ k, we define Ni := {σ ⊆ [n] : η(σ) = i}. 2.2. Stanley-Reisner rings and Betti numbers Suppose ([n], Δ) is as in the previous subsection. Then Δ is a simplicial complex. We denote by IΔ the  ideal of the polynomial ring R := Fq [X1 , . . . , Xn ] generated by all monomials of the form i∈τ Xi , where τ ∈ 2[n] \ Δ. The quotient RΔ = R/IΔ is called the Stanley-Reisner ring or the face ring associated to Δ. As noted in the Introduction, RΔ has a minimal free resolution of the form (1). Furthermore, since IΔ is a monomial ideal generated by squarefree monomials, we can choose the free R-modules Fi in (1) to be not only Z-graded as in (2), but also Zn -graded so as to write Fi =



R(−σ)βi,σ

for i = 1, . . . , k.

(4)

σ∈Zn

In fact, the Zn -graded Betti numbers βi,σ have the property that βi,σ = 0 unless the n-tuple σ = (σ1 , . . . , σn ) has all its coordinates in {0, 1}. Such n-tuples in {0, 1}n can be naturally identified with subsets of [n] where (σ1 , . . . , σn ) corresponds to the subset {i ∈ [n] : σi = 1} of [n] that we shall also denote by σ. Thus, we may

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index the direct sum in (4) by σ ∈ 2[n] . The relation between the Z-graded and Zn -graded Betti numbers is simply that βi,j =



βi,σ

for i = 1, . . . , k.

(5)

|σ|=j

Johnsen and Verdure [16] proved an important relationship between the Zn -graded Betti numbers and subsets of a given nullity. Namely, for 1 ≤ i ≤ k and σ ⊆ [n], βi,σ = 0 ⇐⇒ σ ∈ Ni and σ is a minimal element of Ni .

(6)

This result, which can perhaps be traced back to Stanley [23, p. 59], will be very useful for us in the sequel. Note also that if μ1 , . . . , μt are squarefree monomials in R which constitute a minimal set of generators of  IΔ and if σj ∈ 2[n] denotes the support of μj (so that μj = i∈σj Xi ) for 1 ≤ j ≤ t, then without loss of generality, we can take the first free R-module in (4) to be F1 =

t 

R(−σj ).

(7)

j=1

Finally, we recall the following general result, which was alluded to in the Introduction. A proof can be found in [5]. We note that a graded module M over a polynomial ring R having projective dimension k will have a minimal free resolution such as (1) with RΔ replaced by M , except in this case F0 may not  be equal to R. In general, we let βi := rankR (Fi ) = j βi,j . Note that if M has a pure resolution of type (d0 , d1 , . . . , dk ), then βi := βi,di for i = 0, 1, . . . , k. Theorem 2.1 (Boij-Söderberg). Let R be the polynomial ring over a field and let M be a graded R-module of finite projective dimension k. Then M is Cohen-Macaulay if and only if its graded Betti numbers satisfy the equations k  

(−1)i j  βi,j = 0

for  = 0, . . . , k − 1.

(8)

i=0 j∈Z

In particular, if the minimal free resolution of M is pure of type (d0 , d1 , . . . , dk ), then (8) implies the Herzog-Kühl formula [14]:       d j   βi = β0   (d − d ) i  j=i j

for i = 1, . . . , k,

(9)

As noted in the Introduction, Stanley-Reisner rings associated to linear codes (or more generally, simplicial complexes corresponding to matroids) are Cohen-Macaulay, and hence the above theorem is applicable; moreover, in this case, β0 = 1. If a [n, k]q -code C has a pure resolution of type (d0 , . . . , dk ), then d0 = 0 and for 1 ≤ i ≤ k, di is precisely the i-th generalized Hamming weight of C, thanks to (3); we will refer to βi = βi,di as the Betti numbers of C in this case. 3. Pure resolution of linear codes In this section we will give a characterization of the purity of the resolution of the Stanley-Reisner ring associated to a linear code in terms of the support weight of certain subcodes of the code. We will then outline some simple applications.

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Let C be a [n, k]q -code and let H = [H1 . . . Hn ] be a parity check matrix of C, where, as before, Hi denotes the ith column of H. For any subset σ of [n], define S(σ) to be the subspace Hi : i ∈ σ of Fqn−k spanned by the columns of H indexed by σ. Note that r(σ) = dim S(σ). Let us also define a related subspace of Fqn by  S(σ) := {x = (x1 , . . . , xn ) ∈ Fqn : xi = 0 for i ∈ / σ and



xi Hi = 0}.

i∈σ

Recall that for any subcode D of C, i.e., a subspace D of C, the support of D is the set Supp(D) of all i ∈ [n] for which there is x = (x1 , . . . , xn ) ∈ D with xi = 0; further, we let wt(D) := |Supp(D)|, and call this the weight of D.   Lemma 3.1. Let σ ⊆ [n]. Then S(σ) is a subcode of C and Supp(S(σ)) ⊆ σ. Proof. Since C = {x = (x1 , . . . , xn ) ∈ Fqn :  inclusion Supp(S(σ)) ⊆ σ is obvious. 

n i=1

 is a subcode of C. The xi Hi = 0}, it is clear that S(σ)

For any σ ⊆ [n], let Fqσ denote the set of all ordered |σ|-tuples (xi )i∈σ of elements of Fq indexed by σ. Consider the map φσ : Fqσ → S(σ) defined by φσ (x) =



xi H i .

(10)

i∈σ

Clearly φσ is a surjective Fq -linear map.  Lemma 3.2. Let σ ⊆ [n] and let φσ be as in (10). Then ker φσ is isomorphic (as a Fq -vector space) to S(σ). Consequently,  dim S(σ) = |σ| − dim S(σ).

(11)

Proof. Consider the map ψ : Fqσ −→ Fqn given by ψ(x) = (v1 , v2 , . . . , vn ), where vi =

xi

if i ∈ σ,

0

otherwise.

 It is easily seen that the restriction of ψ to ker φσ gives an isomorphism of ker φσ onto S(σ). The second assertion follows from the Rank-Nullity theorem.  For 0 ≤ i ≤ k, let Gi (C) denote the Grassmannian of all i-dimensional subspaces of C. We call D ∈ Gi (C) an i-minimal subcode of C if Supp(D) is minimal among the supports of all i-dimensional subcodes of C, i.e., Supp(D )  Supp(D) for any D ∈ Gi (C) with D = D. We let Di = the set of all i-minimal subcodes of C. Note that if i = 0, then the only element of Gi (C) is {0}, and its support is ∅, which is clearly i-minimal. Moreover, r(∅) = 0 = |∅|, and thus Supp({0}) ∈ N0 . In fact, a more general result holds. Recall (from § 2.1) that Ni denotes the set of all subsets of [n] of nullity i. Proposition 3.3. Suppose 0 ≤ i ≤ k and D ∈ Di . Then Supp(D) ∈ Ni .

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Proof. Let σ := Supp(D). Then for any x ∈ D, clearly xi = 0 for all i ∈ [n] with i ∈ / σ. Also, since D ⊆ C,    we see that xi Hi = 0 for each x = (x1 , . . . , xn ) ∈ D. It follows that D ⊆ S(σ). In particular, dim S(σ) ≥ i. Further, by Lemma 3.1,  σ = supp(D) ⊆ supp(S(σ)) ⊆ σ.    Therefore supp(S(σ)) = σ. In case dim(S(σ)) > i, we can choose some j ∈ σ and observe that {x ∈ S(σ) :  xj = 0} is a subspace of dimension dim S(σ) − 1, and its support is contained in σ \ {j}. This can be used to  construct an i-dimensional subcode D of S(σ) with support a proper subset of σ. But then the minimality   of the support of D is contradicted. It follows that dim S(σ) = i, and hence D = S(σ). Now equation (11) shows that r(σ) = |σ| − i, that is, σ ∈ Ni .  It turns out that a partial converse of the above proposition is also true. Proposition 3.4. Suppose 0 ≤ i ≤ k and σ is a minimal element of Ni (with respect to inclusion). Then there exists D ∈ Di such that σ = Supp(D).  Proof. Since σ ∈ Ni , we see that dim S(σ) = r(σ) = |σ| − i. Hence, equation (11) implies that dim S(σ) = i.   Let D := S(σ) and σ := Supp(D). Then D is an i-dimensional subcode of C and by Lemma 3.1, σ  ⊆ σ. We claim that D ∈ Di . To see this, assume the contrary. Then there exists D ∈ Gi (C) with D = D such that Supp(D )  σ  . Replacing D by an i-dimensional subcode with smaller support, if necessary, we may assume that D is i-minimal. But then by Proposition 3.3, Supp(D ) ∈ Ni , which contradicts the minimality of σ in Ni . Thus, D ∈ Di .  Corollary 3.5. Suppose 0 ≤ i ≤ k and σ ⊆ [n]. Then σ is a minimal element of Ni if and only if there exists an i-minimal subcode D of C with Supp(D) = σ. Proof. Follows from Propositions 3.3 and 3.4.  Theorem 3.6. Let C be an [n, k]q code and d1 < · · · < dk its generalized Hamming weights. Then any Ngraded minimal free resolution of C is pure if and only if for each i = 1, . . . , k, all the i-minimal subcodes of C have support weight di . Proof. From (6) and Corollary 3.5, we see that for 1 ≤ i ≤ k and σ ⊆ [n], βi,σ = 0 ⇐⇒ σ = Supp(D) for some D ∈ Di .

(12)

Thus, the desired result follows from (3) and (5).  Corollary 3.7. The Betti numbers at the first step of a [n, k]q -code C are given by β1,j = |{D ∈ D1 : wt(D) = j}| for any nonnegative integer j. Proof. Follows from (7) and (12).  Remark 3.8. Let C be an [n, k]q code and h a positive integer ≤ k. Given a resolution of C, say (1), by its left part after h steps, we mean the exact sequence Fk −→ Fk−1 −→ · · · −→ Fh

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which is a minimal free resolution of the cokernel of the last map Fh+1 −→ Fh . Now let d1 < · · · < dk be the generalized Hamming weights of C. It is clear that the proof of Theorem 3.6 also shows that the left part after h steps of any N-graded minimal free resolution of C is pure if and only if for each i = h, . . . , k, all the i-minimal subcodes of C have support weight di . We now show how a characterization due to Johnsen and Verdure [16] of MDS codes can be deduced from our characterization of purity, and moreover, how the minimal free resolution of an MDS code can then be readily determined using the Herzog-Kühl formula. Corollary 3.9. Let C be a nondegenerate [n, k]q -code and h a positive integer ≤ k. Then C is h-MDS if and only if the left part of its resolution after h steps is linear. In particular, C is an MDS code if and only if its resolution is linear. Moreover, if C is MDS, then its Betti numbers are given by

βi =



 n−k+i−1 n i−1 k−i

for i = 1, . . . , k.

Proof. Suppose the left part after h steps of a resolution of C is linear. Since C is nondegenerate, dk = n, and so from the linearity together with equation (3), we obtain di = n − k + i for h ≤ i ≤ k. Taking i = h, we see that C is h-MDS. Conversely, suppose C is h-MDS. Then from the strict monotonicity of generalized Hamming weights [25, Thm. 1], we see that di = n − k + i for h ≤ i ≤ k. Now fix i ∈ {h, . . . , k} and let D be an i-minimal subcode of C. Let σ := Supp(D). By Proposition 3.3, σ ∈ Ni . Also, n − k + i = di ≤ |σ|. Consequently, n − k ≤ |σ| − i = r(σ) ≤ n − k. It follows that |Supp(D)| = di . Thus, in view of Remark 3.8, we conclude that the left part after h steps of any resolution of C is linear. Now assume that C is MDS. Then, in view of (9), we see that for 1 ≤ i ≤ k,

βi =

 j=i

n−k+j dj = = |dj − di | |j − i|

i−1 

j=i

and an easy calculation shows that this is equal to

n−k+j i−j j=1

n−k+i−1 i−1

n k−i

.

   k n−k+j , j−i j=i+1 

Let us also show how the result of Johnsen and Verdure [17] about the minimal free resolution of constant weight codes can be deduced from Theorem 3.6. Corollary 3.10. Let C be an [n, k]q -code in which each nonzero codeword has constant weight d. Then the N-graded resolution of C is pure. Moreover, the generalized Hamming weights (or the shifts) and the Betti numbers of C are given by q k−1 (q i − 1) di = i−1 q (q − 1) where

k i q

and

  i(i−1) k βi = q 2 , i q

for i = 1, . . . , k,

denotes the Gaussian binomial coefficient.

Proof. It is well-known (see, e.g., [20, Thm. 1]) that every j-dimensional subcode of the constant weight code C has support weight dj , where dj =

d(q j − 1) q j−1 (q − 1)

for j = 1, . . . , k.

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Hence, by Theorem 3.6, C has a pure resolution. Evidently, the numbers di defined above are the generalized Hamming weights of C. Moreover, for i, j = 1, . . . , k, d(q i−j − 1) , q i−1 (q − 1)

di − dj =

if j < i, whereas dj − di =

d(q j−i − 1) , q j−1 (q − 1)

if j > i.

Hence, the Herzog-Kühl formula (9) implies that for i = 1, . . . , k, βi =

 j=i

dj = |dj − di |

i−1 

q i−j (q j − 1) q i−j − 1 j=1

  k

qj − 1 q j−i − 1 j=i+1

 =q

i(i−1) 2

  k , i q

where the last equality follows by noting that for i = 1, . . . , k,     k  k k (q k − 1)(q k−1 − 1) · · · (q i+1 − 1) qj − 1 = = k−i = . i q k−i q (q − 1)(q k−i−1 − 1) · · · (q − 1) j=i+1 q j−i − 1 This proves the desired result.  4. Reed-Muller codes In this section we consider generalized Reed-Muller codes and prove that the resolution of the first order Reed-Muller code is pure, whereas for other Reed-Muller codes, it is non-pure. Let us begin by recalling the construction of (generalized) Reed-Muller codes. Fix integers r, m such that m ≥ 1 and 0 ≤ r ≤ m(q − 1). Define Vq (r, m) = {f ∈ Fq [X1 , . . . , Xm ] : deg f ≤ r and degXi f < q for i = 1, . . . , m}. Fix an ordering P1 , . . . , Pqm of the elements of Fqm . Consider the evaluation map Ev : Vq (r, m) → Fqq

m

defined by f → cf := (f (P1 ), . . . , f (Pqm )) .

The image of Ev is called the generalized Reed-Muller code of order r, and we denote it by RMq (r, m). It is well-known that RMq (r, m) is an [n, k, d]q -code, with m

n=q ,



 m  m + r − iq i m , k= (−1) i m i=0

and d = (q − s)q m−t−1 ,

(13)

where t, s are unique integers satisfying r = t(q −1) +s and 0 ≤ s ≤ q −2. Further, for any ω0 , ω1 , . . . , ωt ∈ Fq with ω0 = 0 and any distinct ω1 , . . . , ωs ∈ Fq , the polynomial f (X1 . . . , Xm ) = ω0

t  i=1

(1 − (Xi − ωi )q−1 )

s 

(Xt+1 − ωj )

j=1

is in Vq (r, m) and Ev(f ) is a minimum weight codeword of RMq (r, m). Moreover, up to a (nonhomogeneous) linear substitution in X1 , . . . , Xm , every minimum weight codeword of RMq (r, m) is of this form; see, e.g., Theorems 2.6.2 and 2.6.3 of [10]. It is also well-known (see, e.g., [2, §5.4]) that the dual of RMq (r, m) is given by3 3 Strictly speaking, for the formula (14) to be valid, we should note that the definition of RMq (r, m) is meaningful also when r = −1 in which case it is the zero code of length q m .

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RMq (r, m)⊥ = RMq (r⊥ , m) where r⊥ + r + 1 = m(q − 1).

9

(14)

In particular, if r = m(q − 1) − 1, then RMq (r, m) is a MDS code (being the dual of RMq (0, m), which is the 1-dimensional code of length q m generated by the all-1 vector). Also if r = m(q − 1), then RMq (r, m) is a MDS code, being the full space Fqm . Finally, if m = 1, then RMq (r, m) is a Reed-Solomon code, and in particular, a MDS code. Thus, in all these “trivial cases”, RMq (r, m) has a pure, and in fact, linear, resolution. The following result deals with the first nontrivial case of r = 1. Theorem 4.1. The N-graded minimal free resolution of the first order Reed-Muller code RMq (1, m) is pure and is given by R(−dm+1 )βm+1 −→R(−dm )βm −→ · · · −→ R(−d1 )β1 −→ R where di = q m − q m−i  for 1 ≤ i ≤ m + 1, and

βi =

⎧ m−i  q m+1−j − 1 ⎪ ⎪ (i+1 2 ) ⎪ q ⎪ ⎨ q m+1−i−j − 1

if 1 ≤ i ≤ m,

j=1

m  ⎪ ⎪ j ⎪ ⎪ ⎩ (q − 1)

if i = m + 1.

j=1

Proof. First, note that dim RMq (1, m) = m + 1. Let i be a positive integer ≤ m + 1. If i = m + 1, then the only i-dimensional subcode of RMq (1, m) is RMq (1, m) itself, and this has support weight q m . Now suppose 1 ≤ i ≤ m. Let D be a subcode of RMq (1, m) of dimension i. Then the support weight of D is clearly q m − |Z(f1 , . . . , fi )|, where f1 , . . . , fi ∈ Vq (1, m) are linearly independent polynomials whose images under Ev form a basis of D, and where Z(f1 , . . . , fi ) denotes the set of common zeros in Fqm of f1 , . . . , fi . Now f1 = · · · = fi = 0 is a system of i linearly independent (not necessarily homogeneous) linear equations in m variables, and thus it has either no solutions (when the system is inconsistent) or exactly q m−i solutions (when the system is consistent). Accordingly, the support weight of D is either q m or q m − q m−i . Moreover, if the former holds, then Supp(D) = {1, . . . , q m }, and so D cannot be an i-minimal subcode of RMq (1, m). It follows that all i-minimal subcodes of RMq (1, m) have the same support weight di = q m − q m−i  for 1 ≤ i ≤ m + 1. Thus, by Theorem 3.6, RMq (1, m) has a pure resolution. Consequently, the Betti numbers of RMq (1, m) can be determined using the Herzog-Kühl formula (9) as follows.

βm+1 =

m 

m m   dj q m − q m−j = = (q j − 1), m−j d − d q m+1 j j=1 j=1 j=1

whereas for 1 ≤ i ≤ m, βi =

dm+1 dm+1 − di

= qi

m 



dj  dj d − di jj>i j

i−1 q m−j (q j − 1)  q m−j (q j − 1) q m−j (q j−i − 1) j=1 q m−i (q i−j − 1) j=i+1

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=q

i(i+1) 2

m 

m−i  (q m+1−j − 1) (q j − 1) (i+1 2 ) = q . (q j−i − 1) (q m+1−i−j − 1) j=i+1 j=1

This proves the theorem.  Remark 4.2. Observe that the pure resolution of RMq (1, m) in Theorem 4.1 is linear only when either m = 1 or m = 2 = q. As noted earlier, RMq (1, m) is a MDS code in this case. Next, we shall show that the minimal free N-resolutions of many generalized Reed-Muller codes of order higher than one are not pure. It will be convenient to consider various cases separately. As usual, we shall say that an element c of a linear code C is a minimal codeword if either c = 0, or if c = 0 and the support of the 1-dimensional subspace c of C spanned by c is minimal among the supports of all 1-dimensional subcodes of C. Evidently, a codeword of minimum weight is minimal, but the converse may not be true. 4.1. Binary case In this subsection we consider the binary case, i.e., when q = 2. We will use the following simple, but useful, observation. It is stated, for instance, in Ashikhmin and Barg [3, Lemma 2.1]. The proof is obvious and is omitted. Lemma 4.3. Let C be a binary linear code and let d = d(C) be its minimum distance. If c ∈ C is not a minimal weight codeword, then c = c1 + c2 for some nonzero c1 , c2 ∈ C such that Supp( c1 ) and Supp( c2 ) are disjoint and Supp( ci )  Supp( c ) for i = 1, 2. In particular, if c ∈ C has wt(c) < 2d, then c is a minimal codeword of C. The following result shows that all “nontrivial” binary Reed-Muller codes of order greater than 1 have a non-pure resolution. Proposition 4.4. Assume that m ≥ 4 and 1 < r ≤ m − 2. Then any minimal free N-resolution of the binary Reed-Muller code RM2 (r, m) is not pure. Proof. The minimum distance of RM2 (r, m) is d := 2m−r and if we let Q(X1 , . . . , Xm ) = X1 X2 · · · Xr−2 (Xr−1 Xr + Xr+1 Xr+2 ), then clearly, Q ∈ V2 (r, m). Moreover, the corresponding codeword cQ = Ev(Q) has weight 6 × 2m−r−2 = 3d/2. Indeed, Q(a1 , . . . , am ) = 0 for (a1 , . . . , am ) ∈ F2m precisely when a1 = · · · = ar−2 = 1, (ar−1 , ar , ar+1 , ar+2 ) is one among (0, 1, 1, 1), (1, 0, 1, 1), (0, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0), and (1, 1, 0, 0), while ar+3 , . . . , am ∈ F2 are arbitrary. Hence, by Lemma 4.3, cQ is a minimal codeword, but it is clearly not of minimum weight. Thus, the desired result follows from Theorem 3.6.  Remark 4.5. As Alexander Barg has pointed out to one of us, the last assertion in Lemma 4.3 can be extended to the q-ary case to show that codewords of weight less than dq/(q − 1) are minimal in C, where C is a q-ary linear code with minimum distance d. However, for q > 2, this is often a restrictive hypothesis, and in the next subsections, we will deal with q-ary Reed-Muller codes using a different strategy. 4.2. The case of t = 0 Let t, s be as in (13) so that r = t(q − 1) + s and 0 ≤ s < q − 1. We will consider the case of Reed-Muller codes of order r > 1 for which t = 0 (so that r = s). Note that such codes are necessarily non-binary, and

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in fact, q ≥ 4. We shall also exclude the case when m = 1, since RMq (r, 1) is a Reed-Solomon (and hence MDS) code for 1 ≤ r ≤ (q − 1). Proposition 4.6. Assume that m ≥ 2 and 1 < r < q − 1. Then any minimal free N-resolution of the Reed-Muller code RMq (r, m) is not pure. Proof. Choose distinct elements ω1 , . . . , ωr−1 ∈ Fq and an arbitrary ω ∈ Fq . Define

Q(X1 , . . . , Xm ) = (X2 − ω)

r−1 

(X1 − ωi ).

i=1

Clearly, Q ∈ Vq (r, m) and the corresponding codeword cQ = Ev(Q) has weight (q − r + 1)(q − 1)q m−2 . On the other hand, by (13), the minimum distance of RMq (r, m) is (q − r)q m−1 . Observe that (q − r + 1)(q − 1)q m−2 − (q − r)q m−1 = (r − 1)q m−2 > 0 since r > 1. It follows that cQ is not a minimum weight codeword. If cQ is a minimal codeword, then Theorem 3.6 implies the desired result. Now suppose cQ is not a minimal codeword of RMq (r, m). Then we can find F ∈ Vq (r, m) such that cF is a minimal codeword of RMq (r, m) and Supp(cF ) ⊂ Supp(cQ ). Again, if cF is not a minimal codeword of RMq (r, m), then we are done. Otherwise, by the characterization of minimum weight codewords of RMq (r, m), we must have

F (X1 , . . . , Xm ) =

r 

(L − ωj )

j=1

for some distinct elements ω1 , . . . , ωr ∈ Fq and some nonzero linear polynomial L in Fq [X1 , . . . , Xm ] that we can assume to be homogeneous (by adjusting ωj , if necessary). Write L = a1 X1 + · · · + am Xm . Since Supp(cF ) ⊂ Supp(cQ ), it follows that L vanishes whenever we substitute X1 = ωi for some i ∈ {1, . . . , r} or we substitute X2 = ω. In particular, a1 ω1 + a2 X2 + · · · + am Xm = ωj for some j ∈ {1, . . . , r}. Comparing the degree in each of the variables X2 , . . . , Xm , we obtain a2 = · · · = am = 0 so that L = a1 X1 . But then L does not vanish when we substitute X2 = ω, and we obtain a contradiction. This proves the proposition.  4.3. The case of 0 < t < m − 1 and 1 < s < q − 1 The arguments here will be similar to those in the previous subsection, except that we have to deal with an additional factor of degree t(q − 1). Note that 1 < s < q − 1 implies that q ≥ 4. Proposition 4.7. Assume that 1 < r < m(q − 1) and moreover, r = t(q − 1) + s with 0 < t < m − 1 and 1 < s < q − 1. Then any minimal free N-resolution of the Reed-Muller code RMq (r, m) is not pure. Proof. Choose distinct elements ω1 , . . . , ωs−1 ∈ Fq and an arbitrary ω ∈ Fq . Define  Q(X1 , . . . , Xm ) =

t  i=1

⎞  ⎛s−1  (Xiq−1 − 1) ⎝ (Xt+1 − ωj )⎠ (Xt+2 − ω) j=1

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Clearly, Q ∈ Vq (r, m) and the corresponding codeword cQ = Ev(Q) has weight (q − s + 1)(q − 1)q m−t−2 . On the other hand, by (13), the minimum distance of RMq (r, m) is (q − s)q m−t−1 . Observe that (q − s + 1)(q − 1)q m−t−2 − (q − s)q m−t−1 = (s − 1)q m−t−2 > 0 since s > 1. Thus, as in the proof of Proposition 4.6, it suffices to show that if there exists F in Vq (r, m) such that cF is a minimum weight codeword with Supp(cF ) ⊂ Supp(cQ ), then we arrive at a contradiction. Again, any such F has to be of the form ⎞  t ⎛ s  q−1  F (X1 , . . . , Xm ) = (Li − 1) ⎝ (Lt+1 − ωj )⎠ i=1

j=1

for some distinct ω1 , . . . , ωs ∈ Fq , and linearly independent linear polynomials L1 , . . . , Lt+1 ∈ Fq [X1 , . . . , Xm ] with Lt+1 homogeneous. Note that Supp(cQ ) is contained in the linear space A = {(a1 , . . . , am ) ∈ Fqm : ai = 0 for i = 1, . . . , t}, which can be identified with Am−t , while Supp(cF ) is contained in the affine space A := {a ∈ Fqm : Li (a) = 0 for i = 1, . . . , t} of dimension m − t. Further, since Supp(cF ) ⊂ Supp(cQ ), we obtain Supp(cF ) ⊆ A ∩ A . Now if A = A , then dim(A ∩ A ) ≤ m − t − 1, and so (q − s)q m−t−1 ≤ q m−t−1 , which is impossible because s < q − 1. This shows that A = A . Consequently, F (0, . . . , 0, Xt+1 , . . . , Xm ) =

s 

Lt+1 (0, . . . , 0, Xt+1 , . . . , Xm ) − ωj

j=1

gives a minimum weight codeword in RMq (s, m − t) whose support contains the support of the codeword of RMq (s, m − t) associated to Q(0, . . . , 0, Xt+1 , . . . , Xm ). But then this leads to a contradiction exactly as in the proof of Proposition 4.6.  4.4. The case of s = 0 Since the binary case and the case t = 0 have already been dealt with in subsections 4.1 and 4.2, we shall assume that q ≥ 3 and 1 ≤ t ≤ m − 1. Then s = 0 implies that r = t(q − 1) > 1. Proposition 4.8. Assume that q ≥ 3 and r = t(q −1) with 1 ≤ t ≤ m −1. Then any minimal free N-resolution of the Reed-Muller code RMq (r, m) is not pure. Proof. Write Fq = {ω1 , . . . , ωq } and pick any ω ∈ Fq . Consider

Q(X1 , . . . , Xm ) =

t−1  i=1

⎛ (Xiq−1 − 1) ⎝

q 

⎞ (Xt+1 − ωj )⎠ (Xt+2 − ω)

j=3

Then deg Q = (t − 1)(q − 1) + (q − 2) + 1 = t(q − 1) = r and so Q ∈ Vq (r, m). Also, we can write Supp(cQ ) = A1 ∪ A2 , where for i = 1, 2, Ai := {a = (a1 , . . . , am ) ∈ Fqm : a1 = · · · = at = 0, at+1 = ωi , and at+2 = ω}. Clearly, A1 , A2 are disjoint and so wt(cQ ) = 2(q − 1)q m−t−1 . The minimum distance of RMq (r, m) in this case is q m−t , and 2(q − 1)q m−t−1 > q m−t , since q ≥ 3. Thus, cQ is not a minimum weight codeword. As in the proof of Proposition 4.6, it suffices to show that the existence of F ∈ Vq (r, m) such that cF is a minimum weight codeword with Supp(cF ) ⊂ Supp(cQ ) leads to a contradiction. By the characterization of minimum

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13

t weight codewords of RMq (r, m), any such F has to be of the form F (X1 , . . . , Xm ) = i=1 (Lq−1 −1) for some i linearly independent linear polynomials L1 , . . . , Lt in Fq [X1 , . . . , Xm ]. Hence, Supp(cF ) is the affine space A := {a ∈ Fqm : Li (a) = 0 for i = 1, . . . , t}. Since Supp(cF ) ⊂ Supp(cQ ), we can argue as in the proof of Proposition 4.7 to deduce that A is in fact, the linear space {a ∈ Fqm : a1 = · · · = at = 0}. We now claim that Supp(cF ) is either disjoint from A1 or from A2 . Indeed, if this is not the case then there are Pi ∈ Supp(cF )∩Ai for i = 1, 2. But then Pλ := P1 + λ(P2 − P1 ) ∈ Supp(cF ) for any λ ∈ Fq , since Supp(cF ) = A is linear. Also since q = 3, we can pick λ ∈ Fq such that λ = 0 and λ = 1. Now Supp(cF ) ⊂ Supp(cQ ) = A1 ∪ A2 leads to a contradiction since the tth coordinate of Pλ is neither ω1 nor ω2 . This proves the claim. It follows that A = Supp(cF ) ⊂ Ai for some i ∈ {1, 2}. But then q m−t ≤ (q − 1)q m−t−1 , which is a contradiction. This proves the proposition.  4.5. The case of t = m − 1 and 1 < s < q − 2 We will now consider the last case of nontrivial Reed-Muller codes RMq (r, m) of order r = t(q − 1) + s, where r > 1 and s = 1, namely, when t = m − 1 and s > 1. Note that if we allow s = q − 2, then RMq (r, m) becomes a MDS code, and so we shall assume that 1 < s < q − 2. In particular, this implies that q ≥ 5. Proposition 4.9. Assume that r = (m − 1)(q − 1) + s with 1 < s < q − 2. Then any minimal free N-resolution of the Reed-Muller code RMq (r, m) is not pure. Proof. As in the proof of Proposition 4.8, write Fq = {ω1 , . . . , ωq } and pick any ω ∈ Fq . Also let ν1 , . . . , νs+1 be any distinct elements of Fq . Consider

Q(X1 , . . . , Xm ) =

m−2 

⎞⎛ ⎞ ⎛ q s+1   (Xiq−1 − 1) ⎝ (Xm−1 − ωj )⎠ ⎝ (Xm − νj )⎠ .

i=1

j=3

j=1

Then deg Q = (m − 2)(q − 1) + (q − 2) + (s + 1) = (m − 1)(q − 1) + s = r and so Q ∈ Vq (r, m). Also, wt(cQ ) = 2(q − s − 1) and Supp(cQ ) ⊂ A1 ∪ A2 , where Ai denotes the affine line {a = (a1 , . . . , am ) ∈ Fqm : a1 = · · · = am−2 = 0, am−1 = ωi } for i = 1, 2. The minimum distance of RMq (r, m) in this case is q − s and it is less than 2(q − s − 1), since s < q − 2. As in the proof of Proposition 4.6, it suffices to show that the existence of F ∈ Vq (r, m) such that cF is a minimum weight codeword with Supp(cF ) ⊂ Supp(cQ ) leads to a contradiction. By the characterization of minimum weight codewords of RMq (r, m), any such F has to be of the form F (X1 , . . . , Xm ) =

m−1  i=1

(Lq−1 − 1) i

s 

(Lm − ωj )

j=1

for some linearly independent linear polynomials L1 , . . . , Lm in Fq [X1 , . . . , Xm ] and distinct ω1 , . . . , ωs ∈ Fq . Also, arguing as in the proof of Theorem 4.8, we see that Supp(cF ) is contained in the affine line A := {a ∈ Fqm : Li (a) = 0 for i = 1, . . . , m − 1}. Now if any two points of Supp(cF ) belong to different affine lines A1 and A2 , then Ai ∩ A is nonempty for i = 1, 2 and dimension considerations imply that A1 = A2 = A , which is a contradiction. Hence, the (q − s) points of Supp(cF ) are contained in Supp(cQ ) ∩ Ai for a unique i ∈ {1, 2}. But then q − s ≤ q − s − 1, which is a contradiction. This proves the proposition.  An easy consequence of the above result is that unlike linear resolutions (which correspond to MDS codes), purity of a resolution is not preserved when passing to the dual.

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Corollary 4.10. There exist linear codes C with a pure resolution such that C ⊥ does not have a pure resolution. Proof. By Theorem 4.1, the first order Reed-Muller code RMq (1, m) has a pure resolution. But the dual of RMq (1, m) is RMq ((m − 1)(q − 1) + (q − 3), m) and it does not have a pure resolution, thanks to Proposition 4.9.  We can consolidate the results in subsections 4.1–4.5 to obtain the following. Theorem 4.11. Assume that m ≥ 2 and 1 < r < m(q − 1) − 1. Write r = t(q − 1) + s, where 0 ≤ t ≤ m − 1 and 0 ≤ s < q − 1. Suppose s = 1. Then any minimal free N-resolution of the Reed-Muller code RMq (r, m) is not pure. Proof. Follows from Propositions 4.4, 4.6, 4.7, 4.8, and 4.9.



5. On the purity and resolutions of some two-weight codes This section is devoted to two-weight codes. A linear code C is said to be a two-weight code if there are two distinct positive integers w1 and w2 such that every nonzero codeword of C has weight either w1 or w2 . We will usually take w1 < w2 so that w1 = d1 (C). We have seen in Corollary 3.10 that the resolution of constant weight codes are pure and their Betti numbers are explicitly known. The first order Reed-Muller codes are examples of two-weight codes, and Theorem 4.1 shows that their resolutions are pure and the Betti numbers can be explicitly determined. Thus, it is natural to ask if every two-weight code has pure resolution. In this section we will choose several examples of two-weight codes given by Calderbank and Kantor [8] and see that some of them have pure resolution and others do not. In [8], these codes are referred to by a nomenclature such as RT1, TF1, TF1d , etc., and this is indicated in parenthesis at the beginning of each of the examples considered here. We also compute the Betti numbers of some of the two-weight codes irrespective of whether or not their resolution is pure. The examples of two-weight codes given in [8] are defined geometrically. So before considering them here, we recall a geometric language for codes and translate our characterization of purity (Theorem 3.6) in this language. As before, fix positive integers n, k with k ≤ n and a prime power q. We denote by P k−1 the (k − 1)dimensional projective space over the finite field Fq . A (nondegenerate) [n, k]q projective system is a multiset of n points in P k−1 that do not lie on a hyperplane of P k−1 . Let P be a [n, k]q projective system. For r = 1, . . . , k, the rth generalized Hamming weight, or the rth higher weight of P is defined by dr (P) = n − max{|P ∩ Πr | : Πr linear subspace of P k−1 with codim Πr = r}. Here the “cardinality” |P ∩ Πr | is understood as the sum of multiplicities of points of P that are in Πr . Note that the only linear subspace of codimension k in P k−1 is the empty set, whereas those of codimension k − 1 consist of a single point. Thus, dk (P) = n

and dk−1 (P) = n − 1.

(15)

We can naturally associate a nondegenerate [n, k]q -linear code to P as follows. Choose representatives P1 , . . . , Pn in Fqk corresponding to the n points of P. Let (Fqk )∗ be the dual space of the vector space Fqk . Consider the evaluation map Ev : (Fqk )∗ → Fqn defined by Ev(f ) = (f (P1 ), . . . , f (Pn )).

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The image of Ev is a linear subspace C of Fqn such that dim C = k and C is not contained in a coordinate hyperplane of Fqn . This, then, is the [n, k]q -linear code associated to P. We refer to Tsfasman, Vlăduţ and Nogin [24] for more on projective systems and simply remark that the above association gives rise to a one-to-one correspondence between the equivalence classes of [n, k]q projective systems and nondegenerate [n, k]q -linear codes, which preserves generalized Hamming weights. Also, subcodes of C of dimension r correspond to linear subspaces of P k−1 of codimension r. Thus, we define the support of a linear subspace Πr of P k−1 with codim Πr = r, to be the multiset P \ P ∩ Πr . This corresponds precisely to the support of the corresponding subcode of C. As a consequence, we obtain the following geometric translation of our characterization of purity. Theorem 5.1. Let P ⊆ P k−1 be an [n, k]q projective system and let C be the corresponding [n, k]q -code. The N-graded minimal free resolution of C is pure if and only if for every 1 ≤ r ≤ k − 1 and every linear subspace Πr ⊂ P k−1 of codimension r, there exists a linear subspace H(Πr ) ⊂ P k−1 of codimension r with Πr ∩ P ⊆ H(Πr ) ∩ P and |H(Πr ) ∩ P| = n − dr (P). Proof. Follows from Theorem 3.6.



Corollary 5.2. P ⊆ P k−1 be an [n, k]q projective system and let C be the corresponding linear code. Then the N-graded resolution of C is always pure at the (k − 1)th and kth step. Proof. From (15), we see that C is (k − 1)-MDS. Thus the desired result follows from Corollary 3.9 and Theorem 5.1.  The following definition from [8] is a geometric counterpart of two-weight codes. Definition 5.3. Let h1 , h2 be distinct nonnegative integers. An [n, k]q projective system P is said to be a projective (n, k, h1 , h2 )q system if every hyperplane of P k−1 intersects P either at h1 points or at h2 points (counting multiplicities). Note that if P is a projective (n, k, h1 , h2 )q system, then every nonzero codeword of the corresponding [n, k]q code C is of Hamming weight w1 or w2 , where wi = n − hi for i = 1, 2. Also note that for i = 1, 2, if Awi denotes the number of codewords of C of weight wi , then Awi = (q − 1)νi ,

(16)

where νi denotes the number of hyperplanes Π of P k−1 such that |Π ∩ P| = hi . The factor (q − 1) is due to the fact that the codewords Ev(f ) and Ev(λf ) of C correspond to the same hyperplane in P k−1 for any λ ∈ Fq with λ = 0. We are now ready to discuss several examples from [8] of two-weight codes, and investigate their purity and minimal free resolutions. We use the following notation. pj = pj (q) := |P j (Fq )| =

q j + q j−1 + · · · + q + 1 if j ≥ 0, 0

if j < 0.

Example 5.4 (RT1). Take the base field as Fq2 and let P = P k−1 (Fq2 ). Consider P = P k−1 (Fq ) as a k projective system in P . If Π is a hyperplane in P , then it is given by an equation of the form i=1 zi Xi = 0, where z1 , . . . , zk ∈ Fq2 , not all zero. Fix a Fq -basis {1, θ} of Fq2 and write zi = ai + θbi , where ai , bi ∈ Fq for   ai ci = 0 and bi ci = 0. i = 1, . . . , k. Then P ∩ Π consists of points (c1 : · · · : ck ) ∈ P k−1 (Fq ) satisfying Now if there is λ ∈ Fq such that ai = λbi for all i = 1, . . . , k, or such that bi = λai for all i = 1, . . . , k,

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then P ∩ Π corresponds to a Fq -rational hyperplane in P k−1 (Fq ). Otherwise, it corresponds to a linear subspace of codimension 2 in P k−1 (Fq ). Thus, |P ∩ Π| = pk−2 (q) or pk−3 (q). It follows that the linear code corresponding to P, say C, is a two-weight code of length pk−1 (q) and dimension k over Fq2 . Also, it is clear that as Πr varies over Fq2 -linear subspaces of codimension r in P , the maximum possible value of |P ∩ Πr | is attained when Πr is Fq -rational, and in that case |P ∩ Πr | = pk−1−r (q) for r = 1, . . . , k. It follows that the higher weights of P are given by dr = pk−1 (q) − pk−1−r (q) for r = 1, . . . , k. To determine the purity of the minimal free resolution of C, fix a Fq2 -linear subspace Π of codimension r in P . Let t := dimFq (Π ∩ P). If Π is not Fq -rational, then t < k − 1 − r. Let {f1 , . . . , ft+1 } be a Fq basis of Π ∩ P. Extend this to a linearly independent set {f1 , . . . , ft+1 , . . . , fk−r } ⊂ P. Note that the set {f1 , . . . , fk−r } is linearly independent over Fq2 . (This can be seen, as before, by expressing the coefficients in a linear dependence relation in terms of 1, θ.) Now if H = H(Πr ) is the linear subspace of P spanned by {f1 , . . . , fk−r }, then Π ∩ P ⊂ H ∩ P and |H ∩ P| = n − dr (P). Thus, Theorem 5.1 shows that the N-graded minimal free resolution of C is pure. Moreover, it is of the form 0 → R(−dk )βk → · · · → R(−d2 )β2 → R(−d1 )β1 → R where dr = pk−1 (q) − pk−1−r (q) and βr ’s are given by Herzog-Kühl equation for r = 1, . . . , k. In fact, this is precisely the resolution for constant weight codes given in Corollary 3.10. It may be noted that even though constant weight codes have been characterized by Johnsen and Verdure [17, Thm. 2 and Prop. 4] as those having a resolution as in Corollary 3.10, the code C is not a constant weight code because it is a code over Fq2 , whereas the characterization is for q-ary codes. Remark 5.5. One can similarly consider P = P k−1 (Fq ) ⊆ P k−1 (Fqm ) for any m ≥ 2, and show that the resolution of the linear code corresponding to this projective system is pure and of the form similar to that in Example 5.4 even though this code is not a two-weight code when m > 2. Example 5.6 (TF1). Assume that q is even and consider the projective plane P 2 over Fq . Let P ⊆ P 2 be a hyperoval, i.e., a set of q + 2 distinct points, no three collinear, with the property that if L is a line in P 2 , then |L ∩ P| = 0 or 2. In this case, the corresponding code is an MDS [q + 2, 3]q -code and the resolution of this code is given by Corollary 3.9. Example 5.7 (TF1d ). Suppose q is even and P is the hyperoval in the projective plane P 2 over Fq as in 2 be the dual projective plane. Consider Example 5.6. Let P P = {L : L is a line in P 2 with |L ∩ P| = 2}. 2 and the points of the projective plane P 2 are lines in P 2 . Note also that any two points of ⊆P Note that P 2  Consequently, |P|  = q+2 . Now consider a line in P correspond to a unique line L in P such that L ∈ P. 2 2 , i.e., a point P of P 2 . Counting the intersection of this line with P  corresponds to counting lines L ⊆ P 2 P that pass through P and intersect the hyperoval P in exactly two points. The cardinality of this intersection depends only on whether or not the chosen point P lies on P. More precisely, if P ∈ P, then any line passing through P will intersect the hyperoval P in two points, and there are exactly (q + 1) such lines. On the other hand, if P ∈ / P, then choosing any point Q on P will correspond to a unique line LQ passing through P and Q such that LQ intersects P in another point Q = Q. Further, since each LQ passes through P , the points Q ∈ P corresponding to Q ∈ P are distinct. Since |P| = q + 2, it follows that there are exactly

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q+2 (q + 2)/2 lines of the form LQ . This shows that P is a q+2 projective system, and it 2 , 3, (q + 1), 2 q q+2 corresponds to an [ 2 , 3]q two-weight code with distinct nonzero weights

w1 =

 q+2 q(q + 1) − (q + 1) = 2 2

and w2 =

 q+2 q+2 q(q + 2) − = . 2 2 2

2 that intersect P in (q + 1) points is |P| = q + 2, whereas the number of Also, the number of lines in P 2 2  2  lines in P that intersect P in q+2 2 points is |P \ P| = q − 1. Thus, in view of (16), we see that the weight  is given by spectrum of the two-weight code corresponding to P Aw1 = (q + 2)(q − 1)

and Aw2 = (q 2 − 1)(q − 1).

Furthermore, any hyperplane section of P has to be either of the following two types: (i) a set consisting of lines passing through a fixed P ∈ P and a varying point of P \ {P }, or (ii) a set consisting of lines of the form LQ where Q varies over a suitable subset of P having (q + 2)/2 elements. Now a set of type (ii) has at least two lines and no two lines in this set can intersect in a point of P. Hence a set of type (ii) can never be contained in any set of type (i). It follows that the purity criterion in Theorem 5.1 is violated (for r = 1). Equivalently, every 1-dimensional subcode of C has minimal support, and since C has two distinct nonzero weights d1 = w1 < w2 , we see that the criterion in Theorem 3.6 is violated (for i = 1). Thus, the resolution of C is not pure. Moreover, in view of (5) and (12), we see that the resolution has two twists at the first step, whereas it is pure at the second and third step, thanks to Corollary 5.2. Hence, the resolution of C is of the form R(−d3 )β3,d3 → R(−d2 )β2,d2 → R(−w2 )β1,w2 ⊕ R(−w1 )β1,w1 where w1 , w2 are as before and

d2 =

 q+2 q(q + 3) −3+2= 2 2

and d3 =

 q+2 (q + 1)(q + 2) −3+3= . 2 2

Moreover, from Corollary 3.7, we see that β1,w1 = (q + 2)

and β1,w2 = (q 2 − 1).

To determine the remaining Betti numbers, let us write X1 = β1,w1 , X2 = β1,w2 , Y = β2,d2 , and Z = β3,d3 . Then the Boij-Söderberg equations (8) give the following system of linear equations 1 − (X1 + X2 ) + Y − Z = 0 −w1 X1 − w2 X2 + d2 Y − d3 Z = 0 −w12 X1 − w22 X2 + d22 Y − d23 Z = 0 Putting the values of w1 , w2 , d2 , d3 , X1 and X2 , we obtain Y =  determines the resolution of the code C corresponding to P.

q(q+1)(q+2) 2

and Z =

q 2 (q+1) . 2

This

Example 5.8 (TF2). Assume that q is even with q > 2. Suppose h is an integer such that 1 < h < q and h divides q. Following Denniston [12], a maximal arc in the projective plane P 2 may be defined as a set of points meeting every line in h points or none at all. Let P ⊆ P 2 be a maximal arc consisting of n = 1 + (q + 1)(h − 1) points. It has been shown by Denniston [12] that such maximal arcs exist. Since |L ∩ P| = 0 or h, for any line L in P 2 , we see that the [n, 3]q -code C corresponding to P is a two-weight

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code (cf. [8]) whose nonzero weights are q(h − 1) and n. Since the second weight of C is the length of C, a minimal 1-dimensional subcode of C must be of minimum weight. Hence, by Theorem 3.6, the minimal free resolution of the code C is pure. Thus, in view of Corollary 5.2, we see that the resolution of C is of the form: R(−d3 )β3,d3 → R(−d2 )β2,d2 → R(−d1 )β1,d1 where d1 = q(h − 1), d2 = (q + 1)(h − 1) and d3 = 1 + (q + 1)(h − 1). Using the Herzog-Kühl formula, one can compute the Betti numbers, and they are β1,d1 = (q + 1)2 −

q , h

β2,d2 = qn,

q and β3,d3 = (h − 1)2 (q + 1) . h

2 of P 2 , Example 5.9 (TF2d ). Let q, h, n and P be as in Example 5.8. Consider the dual projective plane P 2 : |L ∩ P| = h}. Now there are exactly (q + 1) lines passing through a point of P 2 , and  = {L ∈ P and let P in case this point is in P, then such a line intersects P in exactly h points. Since |P| = n, it follows that  = n ˆ := |P|

(q + 1)n (q + 1) (1 + (q + 1)(h − 1)) = . h h

2 . Note that a hyperplane, say H,  with a hyperplane of P Next we want to understand the intersection of P 2  2 of P corresponds to a point, say P , of P , and  = {L ⊂ P 2 : L is a line passing through P and |L ∩ P| = h}. H ∩P  is (q +1) or n/h, according as P ∈ P or P ∈  is an (ˆ Therefore |H ∩ P| / P. Thus P n, 3, (q +1), nh )q projective system and the corresponding [ˆ n, 3]q -code is a two-weight code with distinct nonzero weights given by w1 = n ˆ − (q + 1) =

q(q + 1)(h − 1) h

and w2 = n ˆ−

n qn = . h h

Using similar arguments as in Example 5.7, we see that the weight spectrum of this code is given by Aw1 = (q − 1)n

and Aw2 = (q − 1)(q + 1)(q − h + 1),

and also that the resolution of this code is of the form R(−d3 )β3,d3 → R(−d2 )β2,d2 → R(−w2 )β1,w2 ⊕ R(−w1 )β1,w1 where β1,w1 = n = 1 + (q + 1)(h − 1) and β1,w2 = (q + 1)(q − h + 1), and in view of Corollary 5.2, d2 = n ˆ −1 and d3 = n ˆ . As in Example 5.7, using the Boij-Söderberg equations (8) and putting all known values, we obtain β2,d2 =

q(q + 1)(qh + h − q) h

and β3,d3 =

q 2 (q + 1)(h − 1) . 2

We remark that when q > 2, Examples 5.6 and 5.7 are special cases of Examples 5.8 and 5.9, respectively, with h = 2. Example 5.10 (TF3). Assume that q > 2. In the finite projective 3-space P 3 over Fq , an ovoid may be defined as a set of q 2 + 1 points, no three of which are collinear (see, e.g., Dembowski [11, p. 48]). Suppose P is an ovoid in P 3 . Then for any hyperplane H of P 3 , the intersection P ∩ H is an ovoid in H  P 2 , and

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hence using [11, p. 48, §49], we see that |H ∩ P| = 1 or q + 1. Let C be the corresponding linear code. Then C is a two-weight code of length n = q 2 + 1, dimension k = 4, and weights w1 = q(q − 1) and w2 = q 2 . The resolution of this code C is pure. To see this, note that if Π is a hyperplane in P 3 intersecting P at only one point, then there is another hyperplane H with |H ∩ P| = q + 1 and Π ∩ P ⊂ H ∩ P. More precisely, let Π ∩ P = {P } and let Q ∈ P be any point other than P . Take any hyperplane H passing through P and Q. Since |H ∩ P| > 1, we must have |H ∩ P| = q + 1. Further, Π ∩ P ⊂ H ∩ P. It follows that all minimal codewords of C are of minimum weight. Hence, by Corollary 3.7, we see that β1,j = 0 for all j = w1 , i.e., the resolution of C is “pure at the first step”. Next, observe that the maximum possible cardinality of L ∩ P is 2 for any line L in P 3 , and there do exist lines L for which |L ∩ P| = 2. Hence, d2 (C) = n − 2 = q 2 − 1. Consequently, C is a 2-MDS code, and hence by Corollary 3.9, the resolution is linear after the second step. This proves that the resolution of C is pure and is of the form R(−(q 2 + 1))β4,q2 +1 → R(−q 2 )β3,q2 → R(−(q 2 − 1))β2,q2 −1 → R(−q(q − 1))β1,q(q−1) where the Betti numbers can be obtained from Herzog-Kühl formula (9) as follows. β4,q2 +1 = β2,q2 −1 =

q 3 (q−1)2 , 2

β3,q2 = (q − 1)(q 2 − 1)(q 2 + 1),

q 3 (q 2 +1) , 2

and

β1,q(q−1) = q(q 2 + 1).

Example 5.11 (RT3). Assume that k ≥ 3. Consider the quadratic extension Fq2 of Fq and the projective variety Pk−2 ⊂ P k−1 (Fq2 ) defined by the equation X1q+1 + · · · + Xkq+1 = 0. Following Bose and Chakravarti [7], we may refer to Pk−2 as the (nondegenerate) Hermitian variety of dimension k − 2. Let Ck−2 be the [nk , k]q2 -code corresponding to Pk−2 , where nk := |Pk−2 |. We know from [7, Theorem 8.1] that nk =

q k − (−1)k



q k−1 − (−1)k−1 . q2 − 1

(17)

To understand the weights of Ck−2 , first note that since x → xq is an involutory automorphism of Fq2 , every hyperplane of P k−1 (Fq2 ) is given by an equation of the form cq1 X1 + · · · + cqk Xk = 0 for some c = (c1 : · · · : ck ) ∈ P k−1 (Fq2 ); we denote this hyperplane by Hc and call it a tangent hyperplane in case c ∈ Pk−2 (see, e.g., Chakravarti [9, §2]). We remark that Hc and c determine each other. In other words, if c, d ∈ P k−1 (Fq2 ), then: Hc = Hd ⇔ c = d. Now from [9, Theorem 3.1] and from Theorem 7.4 as well as Theorem 8.1 (and its corollary) of [7], we see that |Hc ∩ Pk−2 | =

nk−1

if Hc is not a tangent hyperplane, 2

1 + q nk−2

if Hc is a tangent hyperplane,

(18)

where nk−1 and nk−2 are given by expressions similar to that in (17) with appropriate substitution. Thus, it follows that Ck−2 is a two-weight code. We will now discuss the nature of the resolution of this code when k = 3 and k = 4. First, suppose k = 3. Then P1 is the Hermitian curve consisting of q 3 + 1 points. If L is a line in P 2 (Fq2 ), then by (18), |L ∩ P1 | is either q + 1 or 1, and thus the two nonzero weights of C1 are given by w1 = q(q 2 − 1) and w2 = q 3 . Moreover, if L1 is a tangent line to P1 so that L1 ∩ P1 consists of a single point, say P , then by choosing another point Q of P1 and a line L2 passing through P and Q, we find

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|L2 ∩ P1 | = q + 1 and L1 ∩ P1 ⊂ L2 ∩ P1 . Consequently, every 1-minimal subcode of C1 has support weight w1 = d1 (C1 ). Thus, as in Example 5.10, we can deduce from Corollary 3.7 that the resolution of C1 is “pure at the first step”. This together with Corollary 5.2 shows that the resolution of C1 is pure and it looks like R(−(q 3 + 1))β3,q3 +1 → R(−q 3 )β2,q3 → R(−q(q 2 − 1))β1,q(q2 −1) where the Betti numbers can be obtained from Herzog-Kühl formula (9) as follows. β1,q(q2 −1) = q 2 (q 2 − q + 1), β2,q3 = (q 3 + 1)(q 2 − 1) and β3,q3 +1 = q(q 2 − 1)(q 2 − q + 1). Next, suppose k = 4. Here P2 is the Hermitian surface with (q 2 + 1)(q 3 + 1) points. Further, by (18), a section P2 ∩ Hc of the Hermitian surface by a tangent hyperplane has q 3 + q 2 + 1 points, while a section P2 ∩ Hd by a non-tangent hyperplane has q 3 + 1 points. Moreover, P2 ∩ Hd  P2 ∩ Hc for any c ∈ P2 and d ∈ P 3 (Fq2 ) \ P2 . Indeed, by [9, Theorem 3.1], P2 ∩ Hd is nondegenerate in Hd  P 2 and so the linear span of points in P2 ∩ Hd is Hd . But then P2 ∩ Hd ⊆ P2 ∩ Hc would imply that Hd ⊆ Hc and hence Hd = Hc , which is a contradiction. (Alternatively, if P2 ∩Hd ⊆ P2 ∩Hc , then q 3 +q 2 +1 = |P2 ∩Hd | = |P2 ∩Hd ∩Hc | ≤ |Hd ∩ Hc | = q 2 + 1, which is a contradiction.) At any rate, it follows that C2 is a two-weight code with the nonzero weights w1 = q 5 and w2 = q 5 + q 2 , and moreover, every 1-dimensional subcode of C2 is minimal. Thus, the resolution of C2 has two twists at the first level and by Corollary 3.7, the corresponding Betti numbers are as follows. β1,w1 = |P2 | = (q 2 + 1)(q 3 + 1)

and β1,w2 = |P 3 (Fq2 ) \ P2 | = q 3 (q 2 + 1)(q − 1).

To understand the behavior of the resolution at the second step, we consider 2-dimensional subcodes of C2 and determine which of these are minimal. Equivalently, we consider the sections P2 ∩ L of the Hermitian surface with a line L in P 3 (Fq2 ). It is shown in [7, §10] (see also [9, §5.2]) that |P2 ∩ L| can only take 3 possible values, namely, q 2 + 1, q + 1, or 1. Accordingly, the line L is referred to as a generator, secant line, or tangent line, respectively. It is clear that if L is a tangent line, then there is a non-tangent line L such that P2 ∩ L ⊂ P2 ∩ L . On the other hand, if L is a secant line, then P2 ∩ L ⊂ P2 ∩ L for any generator L , because there is a unique line passing through any two points of P 3 (Fq2 ). It follows that there are two types of 2-minimal subcodes of C2 , one with support weight d2 = |P2 | − (q 2 + 1) = q 3 (q 2 + 1) and another with support weight d2 = |P| − (q + 1) = q(q 4 + q 2 + q − 1). Thus, it follows from (5) and (12) that the resolution of C2 has two twists at level 2, and these correspond to the above values of d2 and d2 . Finally, we note that C2 is 3-MDS and by Corollary 5.2, the resolution of C2 is pure at the third and fourth steps. Thus, we can conclude that the minimal free resolution of C2 has the form R(−d4 )z → R(−d3 )y → R(−d2 )x1 ⊕ R(−d2 )x2 → R(−w2 )β1,w2 ⊕ R(−w1 )β1,w1 where w1 , w2 , d2 , d2 are as before, d3 = (q 2 + 1)(q 3 + 1) − 1, d4 = (q 2 + 1)(q 3 + 1), and x1 , x2 , y, z denote the undetermined Betti numbers, namely, x1 = β2,d2 ,

x2 = β2,d2 ,

y = β3,d3 ,

z = β4,d4 .

To determine these, we note that the Boij-Söderberg equations (8) give rise to 1 − (β1,w1 + β1,w1 ) + (β2,d2 + β2,d2 ) − β3,d3 + β4,d4 = 0 −(w1 β1,w1 + w2 β1,w1 ) + (d2 β2,d2 + d2 β2,d2 ) − d3 β3,d3 + d4 β4,d4 = 0

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−(w12 β1,w1 + w22 β1,w1 ) + (d22 β2,d2 + d22 β2,d2 ) − d23 β3,d3 + d24 β4,d4 = 0 −(w13 β1,w1 + w23 β1,w1 ) + (d32 β2,d2 + d23 β2,d2 ) − d33 β3,d3 + d34 β4,d4 = 0 and this is a system of four linear equation in four unknowns. Substituting the values of the known quantities and solving, we obtain β2,d2 = q 2 (q 3 + 1)(q + 1),

β2,d2 = q 6 (q 2 + 1)(q 2 − q + 1),

β3,d3 = q 3 (q 2 + 1)(q 3 + 1)(q 3 − q + 1), and β2,d4 = q 9 (q 2 − q + 1). Thus, the resolution of C2 is completely determined. We remark that when k ≥ 5, the minimum weight of Ck−2 will be nk − nk−1 or nk − 1 − q 2 nk−2 according as k is odd or even. Moreover, a cardinality argument similar to the one in the case of k = 4 will show that all 1-dimensional subcodes of Ck−2 are minimal, and hence the resolution is not pure (at the first step). It would be interesting to completely determine the resolution of Ck−2 , in general. Acknowledgements We are grateful to Trygve Johnsen for helpful discussions and encouragement. The authors are also grateful to the Indo-Norwegian project MAIT: EECC, supported by the Research Council of Norway and the Dept. of Science and Technology of Govt. of India, which facilitated mutual visits that were helpful to complete this work. The second named author would like to acknowledge past funding, during earlier stages of this work, from the Council of Scientific and Industrial Research, India for a doctoral fellowship at IIT Bombay, and from H.C. Ørsted cofund postdoctoral fellowship at the Technical University of Denmark. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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